1
Lévy-Kolmogorov scaling of turbulence
W. Chen and H.B. Zhou
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Division
Box 26, Beijing 100088, China (Both authors contributed equally to this work,
corresponding to chen_wen@iapcm.ac.cn)
The Kolmogorov scaling law1,2 of turbulences has been considered the most
important theoretical breakthrough in the last century. It is an essential approach
to analyze turbulence data present in meteorological, physical, chemical, biological
and mechanical phenomena3,4. One of its very fundamental assumptions is that
turbulence is a stochastic Gaussian process in small scales5. However, experiment
data at finite Reynolds numbers have observed a clear departure from the
Gaussian5-9. In this study, by replacing the standard Laplacian representation of
dissipation in the Navier-Stokes (NS) equation with the fractional Laplacian10,11,
we obtain the fractional NS equation underlying the Lévy stable distribution which
exhibits a non-Gaussian heavy trail and fractional frequency power law
dissipation12. The dimensional analysis of this equation turns out a new scaling of
turbulences, called the Lévy-Kolmogorov scaling, whose scaling exponent ranges
from -3 to -5/3 corresponding to different Lévy processes and reduces to the
limiting Kolmogorov scaling -5/3 underlying a Gaussian process. The truncated
Lévy process and multi-scaling due to the boundary effect is also discussed.
Finally, we further extend our model to reflecting the history-dependent fractional
Brownian motion.
PACS: 52.35.Ra, 47.53.+n, 92.10.Lq, 05.40.Fb
2
Turbulence occurs all over nature from the atmosphere to the oceans to
electronics to inside stars and internal combustion chambers. Scaling methods are used
to explore hidden structures in the random behavior of turbulent fluid flow even without
a detailed solution of the equations of motion. In the limit of vanishing viscosity (i.e.,
infinite
Reynolds
2
3
E (k ) = Cε k
5
−
3
number),
Kolmogorov’s
celebrated
scaling
of
turbulence
is established1,2, where E(k) is the energy spectrum, C denotes a
absolute constant, and ε represents the kinetic energy dissipation rate and is considered
scale-independent13. In essence, the Kolmogorov -5/3 scaling characterizes the
statistical similarity of turbulent motion at small scales based on the argument of local
homogeneous isotropy
14
. To some extent, the scaling law has been validated by
numerous experimental and numerical data of sufficiently high Reynolds number
turbulence3,4,15. However, recent experiments6,7 using high speed optical techniques
reveal that the statistics of the Lagrangian acceleration manifests distribution profiles
with long heavy tails, indicative of strong non-Gaussian process. This contradicts the
very fundamental foundation of the Kolmogorov theory that turbulence obeys Gaussian
distribution5. Therefore, the -5/3 scaling does not fit real-world turbulences at finite
Reynolds numbers.
It has be suggested that the Lévy β-stable distribution is a proper statistical
approach to accommodate heavy tails widely observed in the probability density
function (PDF) of turbulence quantities5,16, where β is the Lévy stability index and
ranges from 0 to 2. And the Gaussian distribution is its limiting β=2 case17. On the other
hand, the fractional Laplacian is a non-local (integro-differential) and positive definite
operator11,18 underlying the Lévy process in a variety of physical master equations such
as the Fokker-Planck equation17 and the anomalous diffusion equation18-20. By replacing
the Laplacian representation of dissipation process with the fractional Laplacian in the
standard NS equation, this study introduces a fractional NS equation
3
∂u
1
1
β 2
+ u ⋅ ∇u = − ∇p − ~ (− ∆ ) u ,
ρ
∂t
Re
β ∈ (0,2] ,
(1)
~
where u represents velocity vector, Re is the scaled Reynolds number, and the
fractional Laplacian (− ∆ )
β 2
serves as a stochastic driver and guarantees the positive
definiteness of energy dissipation. The essence of fractional NS equation (1) is that the
constitutive equation in turbulence may not obey the classical Newtonian gradient law.
Using a Kolmogorov-like argument, the dimensional analysis of equation (1) leads
to the energy spectrum
2
−
~
E (k ) = Cε~ 3 k
9− 2 β
3
,
(2)
~
where C and ε~ are respectively the scaled C and ε by parameter β. The
corresponding exponent of decay power law ranges from -3 to -5/3, in which the upper
limiting β=2 leads to the Kolmogorov -5/3 scaling corresponding to the Gaussian
distribution and the standard NS equation, while the lower limiting β→0 coincides with
-3 power scaling which has been observed in experiments21. Mandelbrot22 pointed out
that the intermittent (non-Gaussian) property of turbulence calls for a power law of
energy spectrum having exponent -5/3-c (c≥0). But he did not quantitatively clarify his
correction as this study did. Since the scaling (2) generalizes the Kolmogorov scaling
and underlies Lévy statistics, we call it the Lévy–Kolmogorov scaling, which reflects
anomalous transportation of kinetic energy, as evidenced in plasma turbulence23. For
example, the turbulent fluids having viscosity β =2 correspond to the Kolmogorov
scaling, but the boundary layer turbulence is known to have β=1/2, a strong nonGaussian Lévy process12, leading to -8/3 scaling.
Unlike Gaussian process, Lévy process does not have finite moments of second
or higher order. And the truncated Lévy distribution was thus proposed in turbulence
modeling24, in which the long fat tails of algebraic decay of the original Lévy
distribution is truncated and replaced by the corresponding Gaussian distribution of
4
exponential decay, and then the divergent second moments are cured. However, this
truncation is somewhat arbitrary and the truncated Lévy distribution can no longer
underlie the fractional Laplacian in the governing equation. It is noted that the standard
Lévy distribution and fractional Laplacian are defined under infinite domain. However,
the real-world turbulences all have finite Reynolds numbers, namely, the finite size of
turbulence region. From this view, we have to take account of boundary effect into
consideration. The standard definition of the fractional Laplacian under infinite domain
encounters hypersingularity10, which corresponds to the infinite moment of the second
and higher orders of Lévy distribution19,20. Chen11 recently introduced a new definition
of fractional Laplacian under finite domain (equation 17 in ref. 11) which naturally
includes boundary conditions and eliminates hypersingularity. Accordingly, the Lévy
distribution corresponding to the fractional Laplacian of finite domain in the NS
equation (1) is truncated in terms of boundary conditions and has the finite square
moment. It is known that the truncated Lévy distribution gives rise to the intermittency
(finite scaling range) and multifractality (multi-scaling) of turbulence phenomena,
which has long been observed as a rule rather than exception in turbulence8,9.
Multifractality suggests that the Lévy-Kolmogorov scaling exponent is not universal
over all scales. It is noted that the standard dimensional analysis of the NS equation
leading to the scaling power law does not consider the effect of boundary conditions.
Turbulence is observed experimentally and numerically to have anomalous
diffusion23,25,26. The above fractional NS equation (1) can be reduced to the well known
anomalous diffusion equation underlying a time-dependent β-stable Lévy process
(η = 2 β , 0<β<2)16,17. Alternatively, by replacing the fractional Laplacian dissipation
in equation (1) with a mixed operator of the fractional time derivative and the Laplacian
1 ∂µ
∆u , 0≤µ<1, we have a new fractional NS equation underlying the fractional
Re ∂t µ
Brownian motion. The dimensional analysis of this equation leads to a scaling,
5
2
E (k ) = Cˆ εˆ 3 k
−
5− 3 µ
3− µ
,
(3)
whose lower and upper limiting exponents are the Kolmogorov scaling -5/3 and -1,
respectively. Furthermore, the combination of the fractional Laplacian and the fractional
time derivative can be used in the NS equation to represent anomalous dissipation
underlying the statistic paradigms of the fractional Brownian and the Lévy processes,
1
1 ∂µ
∂u
(− ∆ )β 2 u ,
+ u ⋅ ∇u = − ∇ p −
µ
ρ
Re ∂t
∂t
(4)
where β ∈ (0,2] when µ=0; µ ∈ [0,1) when β=2; and Re represents the time-space
scaled Reynolds number. The above NS equation (4) reflects spatial and temporal
fractal irregularity and memory effect of turbulent motions. The energy transfer scaling
of turbulence is thus split into three phases: sub-transport (subdiffusion, fractional
Brownian motion) before Kolmogorov scaling range, normal-transport (Kolmogorov
scaling), and super-transport (Lévy-Kolmogorov scaling, e.g., ballistic transport with
β=1, µ=0 and -7/3 scaling). The orders β and µ of the fractional derivative are
considered time and space fractal dimensions22. Serving as an example, we consider the
elastic turbulence of polymer solutions27, whose equation for motion differs from the
standard NS equation and has to reflect the history-dependent motion. The above NS
equation (4) reflects the constitutive relationship between stress and the fractional time
derivative representation of the deformation rate in a polymer flow and is a competitive
alternative11,12 to the conventional nonlinear constitutive model27. For the case β=2 and
µ≠0, the scaling exponent of polymer fluid turbulence is not the Kolmogorov index -5/3
even in the limit of infinite Reynolds number.
For a further analysis, the NS equation (4) is reduced to the well-known
anomalous diffusion equation18,19
6
∂1− µ u
β
+ γ (− ∆ ) u = 0 ,
1− µ
∂t
(5)
whose corresponding discrete Lagrangian stochastic model is defined by the time
evolution of the mean square displacement of diffusing particle movements
∆x 2 ∝ ∆t η , where η=2(1-µ)/β, ∆x represents distance, ∆t denotes time interval, the
brackets represent the mean value of random variables (e.g., a collection of particles).
Accordingly we can compare our theoretical predictions with experimental data, where
the motion of tracer particles in turbulent flows is measured. For instance, the wellknown
∆x 2 ∝ ∆t 3 in turbulence is first obtained by Richardson28 to explain
experimental measurements and results in µ=0, β=2/3 in the NS equation (4) and -23/9
scaling. It is stressed that the corresponding anomalous diffusion equation (5) of
fractional time-space derivatives is physically more reasonable than the Richardson’s
diffusion equation with a space- and time-dependent diffusion coefficient for deriving
∆x 2 ∝ ∆t 3 , since integer-order differentiability in the latter may not exist in turbulent
velocity flow fields.
It is known that Gaussian process corresponds to the normal diffusion (η = 1 ),
while Lévy process reflects the superdiffusion (long-range spatial correlation, η f 1 )
and the fractional Brownian motion underlies the subdiffusion ( η p 1 ) which manifests
history-dependent (long-range temporal correlation) motion. Once the signature η of
anomalous diffusion is known in turbulence, it is straightforward to derive the order of
fractional derivatives of our NS equation model and then the scaling exponent. For a
subdiffusion case,
∆x 2 ∝ ∆t 0.5 is found in magnetic field turbulence and cosmic-ray
transport in the interstellar medium23,26,29. By using the previous formulas, it is simple
to find µ=0.5, β=2 in NS equation (4) and the corresponding scaling exponent -7/5.
In summary, this study introduced the fractional Laplacian NS equation and then
presented a new Lévy-Kolmogorov -3 to -5/3 scaling of turbulence. The fractional time
7
derivative was also used in the representation of dissipation in the NS equation
underlying the history-dependent fractional Brownian motion and leads to the scaling
law -5/3 to -1. Warhaft30 pointed out “Apart from noting the presence of non-Gaussian
tails, no deeper analysis of the shape of the pdfs has been made. Because the connection
of these models to the Navier-Stokes equations is tenuous,...”. In this study, the first
attempt was made to explicitly connect non-Gaussian statistics of turbulence and the NS
equation, where the fractional derivative representations reflect the strong influence of
the viscose property of fluids on turbulence and vice versa. It is also worth noting that
the direct numerical simulation of the fractional NS equation will be more challenging
than that of the standard NS equation, since the fractional derivatives are a non-local
operator and will result in the full matrix of numerical discretization which is usually
computationally very costly.
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